On the diagonal stability of some classes of complex systems

Abstract

The paper deals with the problem of diagonal stability of nonlinear difference-differential systems. Certain classes of complex systems with delay and nonlinearities of a sector type are studied. It is assumed that these systems describe the interaction of two-dimensional blockswith a delay in connections between the blocks. Two kinds of structure of connections are investigated. For every kind, necessary and sufficient conditions for the existence of diagonal Lyapunov—Krasovskii functionals are found. The existence of such functionals guarantees the asymptotic stability of the zero solutions of considered systems for any nonnegative delay and any admissible nonlinearities. These conditions are formulated in terms of the Hurwitz property of specially constructed Metzler matrices. The proposed approaches are used for the stability analysis ofsome models of population dynamics. Generalized Lotka—Volterra models composed of several interacting pairs of predator-prey type are investigated. With the aid of the Lyapunov direct method and diagonal Lyapunov— Krasovskii functionals, conditions are derived under which equilibrium positions of the considered models are globally asymptotically stable in the positive orthant of the state space for any nonnegative delay. An illustrative example and results of the numerical simulation are presented to demonstrate the effectiveness of the developed approaches.